Introduction to topological insulators and superconductors

古崎　昭　（理化学研究所）

topological insulators (3d and 2d)September 16, 2009

Outline• イントロダクション バンド絶縁体 , トポロジカル絶縁体・超伝

導体• トポロジカル絶縁体・超伝導体の例 : 整数量子ホール系 p+ip 超伝導体 Z2 トポロジカル絶縁体 : 2d & 3d• トポロジカル絶縁体・超伝導体の分類

絶縁体　 Insulator

• 電気を流さない物質

絶縁体

Band theory of electrons in solids

• Schroedinger equation

ion (+ charge)

electron Electrons moving in the lattice of ions

rErrVm

2

2

2

rVarV

Periodic electrostatic potential from ions and other electrons (mean-field)

Bloch’s theorem:

ruarua

ka

ruer knknknikr

,,, , ,

kEn Energy band dispersion :n band index

例：一次元格子

2

A A A A AB B B B B

t

n 1n1n

B

Aikn

u

uen

B

A

B

A

u

uE

u

u

kt

kt

)2/cos(2

)2/cos(2

222 )2/(cos4 ktE

k

E

0

2

Metal and insulator in the band theory

k

E

a

a 0

Interactions between electronsare ignored. (free fermion)

Each state (n,k) can accommodateup to two electrons (up, down spins).

Pauli principle

Band Insulator

k

E

a

a 0

Band gap

All the states in the lower band are completely filled. (2 electrons per unit cell)

Electric current does not flow under (weak) electric field.

2

2

Topological (band) insulators

Condensed-matter realization of domain wall fermions

Examples: integer quantum Hall effect,

quantum spin Hall effect, Z2 topological insulator, ….

•　バンド絶縁体•　トポロジカル数をもつ•　端に gapless 励起 (Dirac fermion) をもつ

stable

free fermions

Topological (band) insulators

Condensed-matter realization of domain wall fermions

Examples:

•　バンド絶縁体•　トポロジカル数をもつ•　端に gapless 励起 (Dirac fermion) をもつ

Topological superconductors

BCS 超伝導体　　　　超伝導 gap

(Dirac or Majorana) をもつ

p+ip superconductor, 3He

stable

Example 1: Integer QHE

Prominent example: quantum Hall effect• Classical Hall effect

B

e

E

v

W

:n electron density

Electric current nevWI

Bc

vE Electric field

Hall voltage Ine

BEWVH

Hall resistancene

BRH

Hall conductanceH

xy R

1BveF

Lorentz force

Integer quantum Hall effect (von Klitzing 1980)

Hxy

xx Quantization of Hall conductance

h

eixy

2

807.258122e

h

exact, robust against disorder etc.

Integer quantum Hall effect• Electrons are confined in a two-dimensional plane. (ex. AlGaAs/GaAs interface)

• Strong magnetic field is applied (perpendicular to the plane)

Landau levels:

,...2,1,0 , ,21 n

mc

eBnE ccn

AlGaAs GaAsB

cyclotron motion

k

E

TKNN number (Thouless-Kohmoto-Nightingale-den Nijs)TKNN (1982); Kohmoto (1985)

Ch

exy

2

Chern number (topological invariant)

yxxy k

u

k

u

k

u

k

urdkd

iC

**22

2

1

yxk kkAkdi

, 2

1 2

filled band

kkkyx uukkA

,

integer valued

)(ruek

rki

Edge states• There is a gapless edge mode along the sample

boundary.B

Number of edge modes Chexy

/2

Robust against disorder (chiral fermions cannot be backscattered)

Bulk: (2+1)d Chern-Simons theoryEdge: (1+1)d CFT

xm

x

Effective field theory

zyyxx xmivH

zyyxx mivH

parity anomaly mxy sgn2

1

Domain wall fermion

i

dxxmv

ikyyxx

y

1''

1exp,

0

vkE 0m 0m

Example 2: chiral p-wave superconductor

BCS 理論　（平均場理論）

,,

*,,

,

22

'','','2

1

2

rrrrrrrrrrdd

rAiem

rrdH

dd

d

''',, rrrrVrr 超伝導秩序変数

S-wave (singlet) rrrrr 2

1'',,

0

P-wave (triplet)

+

rrrr

rr

''

',,

yxkk ikkk 0)(

高温超伝導体 22)( yxkk kkk

22 yxd

Integrating out Ginzburg-Landau

Spinless px+ipy superconductor in 2 dim.• Order parameter

• Chiral (Majorana) edge state

)()( 0 yxkk ikkk

k

E

2

1

2)()(

)(2)(

2

122

22

k

FFyx

FyxF

kh

mkkkikk

kikkmkkH

Hamiltonian density

k

k

m

kkk

k

k

k

kh Fyx

F

y

F

xk 2

222

khE

Gapped fermion spectrum22 : SS

h

h

k

k

winding (wrapping)number=1

1zL

yxyxF

iik

im

H

222

2

rv

rue

tr

tr iEt /

,

,

v

uE

v

u

hii

iih

yx

yx

0

0

Hamiltonian density

Bogoliubov-de Gennes equation

*

*

,u

v

v

uEE Particle-hole symmetry (charge conjugation)

zeromode: Majorana fermion

Ht

i ,

Majorana edge state

v

uE

v

u

km

ik

i

ik

ikm

FyxyxF

yxF

Fyx

222

222

2

12

1

E

px+ipy superconductor: 0y 0

v

u

y

1

1 cosexp 22 ykky

k

mikx

v

uF

Fk

k

Fk

kE

k

E

2

F

FF

k

kktxik

kktxik ee

dktx

0

//

4,

yydyk

ik

kdkH y

F

k

kkF

F

0

edge

• Majorana bound state in a quantum vortex

vortex e

hc

Fn En / , 2000

Bogoliubov-de Gennes equation

v

u

v

u

he

ehi

i

*0

0 FEAepm

h 2

0 2

1

zero mode

v

u

00 00 Majorana (real) fermion!

energy spectrum of vortex bound states

2N vortices GS degeneracy = 2N

interchanging vortices braid groups, non-Abelian statistics

i i+11 ii

ii 1

D.A. Ivanov, PRL (2001)

Fractional quantum Hall effect at • 2nd Landau level• Even denominator (cf. Laughlin states: odd denominator)

• Moore-Read (Pfaffian) state

2

5

jjj iyxz

4/2MR

21Pf iz

jiji

ji

ezzzz

ijij AA det Pf

Pf( ) is equal to the BCS wave function of px+ipy pairing state.

Excitations above the Moore-Read state obey non-Abelian statistics.

Effective field theory: level-2 SU(2) Chern-Simons theory

G. Moore & N. Read (1991); C. Nayak & F. Wilczek (1996)

Example 3: Z2 topological insulator

Quantum spin Hall effect

Quantum spin Hall effect (Z2 top. Insulator)

• Time-reversal invariant band insulator• Strong spin-orbit interaction• Gapless helical edge mode (Kramers pair)

Kane & Mele (2005, 2006); Bernevig & Zhang (2006)

B

B

up-spin electrons

down-spin electrons

SEpSL

,,,spin

,,charge

2

0

xyxyxyxy

xyxyxy

If Sz is conserved, If Sz is NOT conserved,

Chern # (Z) Z2

Quantized spin Hall conductivity

(trivial)Band insulator

Quantum SpinHall state

Quantum Hallstate

ky

kx

E

K

K’

K’

K’

K

K

Kane-Mele model (PRL, 2005)

ij ij i

iiivij

jzijiRjz

iijSOji cccdscicscicctH

t

SOi

SOi

i j

ijd

A B , ,, skke sBsA

rki

zv

yxxyR

zzSOy

yx

xK sssivHK

2

333 :

(B) 1 (A), 1 i

zv

yxxyR

zzSOy

yx

xK sssivHK

2

333 :' '

'*

Ky

Ky HisHis time reversal symmetry

• Quantum spin Hall insulator is characterized by Z2 topological index

1 an odd number of helical edge modes; Z2 topological insulator

an even (0) number of helical edge modes01 0

Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)]

Quantum spin Hall effect

2

esxy

(if Sz is conserved)

0R

Edge states stable against disorder (and interactions)

Z2 topological number

Z2: stability of gapless edge states

(1) A single Kramers doublet

zzyyx

xxz sVsVsVVivsH 0

HisHis yy *

(2) Two Kramers doublets

zzyyx

xy

xz sVsVsVVsIivH 0

Kramers’ theorem

opens a gap

Odd number of Kramers doublet (1)

Even number of Kramers doublet (2)

ExperimentHgTe/(Hg,Cd)Te quantum wells

Konig et al. [Science 318, 766 (2007)]

CdTeHgCdTeCdTe

Example 4: 3-dimensional Z2 topological insulator

3-dimensional Z2 topological insulatorMoore & Balents; Roy; Fu, Kane & Mele (2006, 2007)

bulkinsulator

surface Dirac fermionZ2 topological insulator

bulk: band insulator

surface: an odd number of surface Dirac modes

characterized by Z2 topological numbers

Ex: tight-binding model with SO int. on the diamond lattice [Fu, Kane, & Mele; PRL 98, 106803 (2007)]

trivial insulator

Z2 topologicalinsulator

trivial band insulator: 0 or an even number of surface Dirac modes

Surface Dirac fermions

• A “half” of graphene

• An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go

theorem

ky

kx

E

K

K’

K’

K’

K

K

topologicalinsulator

Experiments• Angle-resolved photoemission spectroscopy (ARPES)

Bi1-xSbx

p, Ephoton

Hsieh et al., Nature 452, 970 (2008)

An odd (5) number of surface Dirac modes were observed.

Experiments II

Bi2Se3

Band calculations (theory)

Xia et al.,Nature Physics 5, 398 (2009)

“hydrogen atom” of top. ins.

a single Dirac cone

ARPES experiment

トポロジカル絶縁体・超伝導体の分類

Schnyder, Ryu, AF, and Ludwig, PRB 78, 195125 (2008)

arXiv:0905.2029 (Landau100)

Classification oftopological insulators/superconductors

Schnyder, Ryu, AF, and Ludwig, PRB (2008)

Kitaev, arXiv:0901.2686

Zoo of topological insulators/superconductors

Topological insulators are stable against (weak) perturbations.

Classification of topological insulators/SCs

Random deformation of Hamiltonian

Natural framework: random matrix theory (Wigner, Dyson, Altland & Zirnbauer)

Assume only basic discrete symmetries:

(1) time-reversal symmetry

HTTH 1* 0 no TRSTRS = +1 TRS with -1 TRS with

TT

TT (integer spin)(half-odd integer spin)

(2) particle-hole symmetry

HCCH 1 0 no PHSPHS = +1 PHS with -1 PHS with

CC

CC (odd parity: p-wave)(even parity: s-wave)

HTCTCH 110133

(3) TRS PHS chiral symmetry [sublattice symmetry (SLS)]

yisT

(2) particle-hole symmetry Bogoliubov-de Gennes

zkyyxxkk

kkkk kkh

c

chccH

2

1px+ipy

Txkxkx CChh *

dx2-y2+idxy

zkyyyxyxkk

kkkk

kkkkhc

chccH

22 2

1

Tykyky CiChh *

10 random matrix ensembles

IQHE

Z2 TPI

px+ipy

Examples of topological insulators in 2 spatial dimensionsInteger quantum Hall EffectZ2 topological insulator (quantum spin Hall effect) also in 3DMoore-Read Pfaffian state (spinless p+ip superconductor)

dx2-y2+idxy

Table of topological insulators in 1, 2, 3 dim.Schnyder, Ryu, Furusaki & Ludwig, PRB (2008)

Examples:(a) Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator,(c) 3d Z2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read),(e) Chiral d-wave superconductor, (f) superconductor,(g) 3He B phase.

Classification of 3d topological insulators/SCs

strategy 　 (bulk boundary)

• Bulk topological invariants　　 integer topological numbers ： 3 random matrix ensembles (AIII, CI,

DIII)

• Classification of 2d Dirac fermions 　 Bernard & LeClair (‘02)

　　 13 classes 　 (13=10+3) AIII, CI, DIII AII, CII

• Anderson delocalization in 2d nonlinear sigma models 　　　 Z2 topological term (2) 　 or 　 WZW term (3)

BZ: Brillouin zone

Topological distinction of ground states

m filledbands

n emptybands

xk

yk

map from BZ to Grassmannian

nUmUnmU2 IQHE (2 dim.)

homotopy class

deformed “Hamiltonian”

In classes AIII, BDI, CII, CI, DIII, Hamiltonian can be made off-diagonal.

Projection operator is also off-diagonal.

nU3 topological insulators labeled by an integer

qqqqqqkd

q

111

2

3

tr24

Discrete symmetries limit possible values of q

Z2 insulators in CII (chiral symplectic)

The integer number # of surface Dirac (Majorana) fermions q

bulkinsulator

surfaceDiracfermion

(3+1)D 4-component Dirac Hamiltonian

mk

kmmkkH x

AII: kHikHi yy *TRS

DIII: kHkH yyyy *PHS

AIII: kHkH yy chS

mq sgn2

1

zm

z

(3+1)D 8-component Dirac Hamiltonian

0

0

D

DH

imkikikD yyy 5CI:

kDkDT

CII: kDmkkD kDikDi yy *

2sgn2

1 mq

0q

zm

z

Classification of 3d topological insulators

strategy 　 (bulk boundary)

• Bulk topological invariants　　 integer topological numbers ： 3 random matrix ensembles (AIII, CI,

DIII)

• Classification of 2d Dirac fermions 　 Bernard & LeClair (‘02)

　　 13 classes 　 (13=10+3) AIII, CI, DIII AII, CII

• Anderson delocalization in 2d nonlinear sigma models 　　　 Z2 topological term (2) 　 or 　 WZW term (3)

Nonlinear sigma approach to Anderson localization• (fermionic) replica• Matrix field Q describing diffusion• Localization massive• Extended or critical massless topological Z2 term

or WZW term

Wegner, Efetov, Larkin, Hikami, ….

22 ZM

ZM 3

Table of topological insulators in 1, 2, 3 dim.Schnyder, Ryu, Furusaki & Ludwig, PRB (2008) arXiv:0905.2029

Examples:(a) Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator,(c) 3d Z2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read),(e) Chiral d-wave superconductor, (f) superconductor,(g) 3He B phase.

Reordered TableKitaev, arXiv:0901.2686

Periodic table of topological insulators

Classification in any dimension

Classification oftopological insulators/superconductors

Schnyder, Ryu, AF, and Ludwig, PRB (2008)

Kitaev, arXiv:0901.2686

Summary• Many topological insulators of non-interacting fermions

have been found. interacting fermions??

• Gapless boundary modes (Dirac or Majorana) stable against any (weak) perturbation disorder

• Majorana fermions to be found experimentally in solid-state devices

Andreev bound states in p-wave superfluids 3He-B

Z2 T.I. + s-wave SC Majorana bound state (Fu & Kane)